Idea

Definition

Of a group on itself

The adjoint action of a groupGG on itself is the actionAd:G×G→GAd : G \times G \to G given by

Ad:(g,h)↦g−1⋅h⋅g.
Ad : (g,h) \mapsto g^{-1} \cdot h \cdot g
\,.

Of a Lie group on its Lie algebra

The adjoint action ad:G×𝔤→𝔤ad : G \times \mathfrak{g} \to \mathfrak{g} of a Lie groupGG on its Lie algebra𝔤\mathfrak{g} is for each g∈Gg \in G the derivativedAd(g):TeG→TeGd Ad(g) : T_e G \to T_e G of this action in the second argument at the neutral element of GG

ad:(g,x)↦Ad(g)*(x).
ad : (g,x) \mapsto Ad(g)_*(x)
\,.

This is often written as ad(g)(x)=g−1xgad(g)(x) = g^{-1} x g even though for a general Lie group the expression on the right is not the product of three factors in any way. But for a matrix Lie groupGG it is: in this case both gg as well as xx are canonically identified with matrices and the expression on the right is the product of these matrices.

Of a Lie algebra on itself

Differentiating the above example also in the second argument, yields the adjoint action of a Lie algebra on itself

ad:𝔤×𝔤→𝔤
ad : \mathfrak{g} \times \mathfrak{g} \to \mathfrak{g}

which is simply the Lie bracket

adx:y↦[x,y].
ad_x : y \mapsto [x,y]
\,.

Of a Hopf algebra on itself

Let kk be a commutative unital ring and H=(H,m,η,Δ,ϵ,S)H = (H,m,\eta,\Delta,\epsilon, S) be a Hopfkk-algebra with multiplication mm, unit map η\eta, comultiplication Δ\Delta, counit ϵ\epsilon and the antipode map S:H→HopS: H\to H^{op}. We can use Sweedler notationΔ(h)=∑h(1)⊗kh(2)\Delta(h) = \sum h_{(1)}\otimes_k h_{(2)}. The adjoint action of HH on HH is given by

h▹g=∑h(1)gS(h(2))
h\triangleright g = \sum h_{(1)} g S(h_{(2)})

and it makes HH not only an HH-module, but in fact a monoid in the monoidal category of HH-modules (usually called HH-module algebra).